Flexible A-scan rate MHz-OCT: efficient computational downscaling by coherent averaging.

In order to realize adjustable A-scan rates of fast optical coherence tomography (OCT) systems, we investigate averaging of OCT image data acquired with a MHz-OCT system based on a Fourier Domain Mode Locked (FDML) laser. Increased system sensitivity and image quality can be achieved with the same system at the cost of lower imaging speed. Effectively, the A-scan rate can be reduced in software by a freely selectable factor. We demonstrate a detailed technical layout of the strategies necessary to achieve efficient coherent averaging. Since there are many new challenges specific to coherent averaging in swept source MHz-OCT, we analyze them point by point and describe the appropriate solutions. We prove that coherent averaging is possible at MHz OCT-speed without special interferometer designs or digital phase stabilization. We find, that in our system up to ∼100x coherent averaging is possible while achieving a sensitivity increase close to the ideal values. This corresponds to a speed reduction from 3.3 MHz to 33 kHz and a sensitivity gain of 20 dB. We show an imaging comparison between coherent and magnitude averaging of a human finger knuckle joint in vivo with 121 dB sensitivity for the coherent case. Further, the benefits of computational downscaling in low sensitivity MHz-OCT systems are analyzed.


Introduction
Modern FDML lasers have paved the way to high fidelity swept source OCT systems with A-scan rates up to several Megahertz [1] and coherence lengths superior to usual spectrometer based systems [2][3][4]. Today also other technologies can achieve MHz A-scan rates [5][6][7][8][9][10]. While MHz-OCT enables many new applications such as wide field ophthalmic [11][12][13][14], video-rate delay-free live 4D-OCT [15,16] and high resolution intravascular OCT without motion artifacts [17,18], the short acquisition times always come at the cost of system sensitivity. At a given laser power incident on the sample, faster systems exhibit lower ultimate shot noise limited sensitivity [19,20]. But in some situations image quality is more important than acquisition speed.
Ideally the same OCT system could perform high speed imaging with reduced sensitivity and high sensitivity imaging at lower speed. Still, the hardware of most high speed swept source OCT systems cannot run at arbitrary A-scan rates, even though there are light sources with adjustable wavelength sweep repetition rate [9,[21][22][23]. This is due to the particular optoelectrical periphery of modern OCT engines that have to be specifically tailored to a certain A-scan rate. In most real OCT systems, analog electronic filters, the bandwidth of photo-receivers, the mechanical response of resonant scanners and several other components must be designed for a certain A-scan rate to achieve shot noise limited sensitivity.
An alternative approach to realize variable A-scan rate OCT is to retain fast image acquisition times, but virtually reduce the A-scan rate afterwards in software by averaging, which also improves sensitivity. The ability of coherent averaging to reduce white Gaussian noise more effectively than incoherent averaging has already been demonstrated on OCT data [24,25]. Coherent averaging has been applied to fast MHz-OCT systems by Siddiqui et al. [10], but not for FDML-lasers.
Only coherent averaging, which comprises averaging of fringes or the complex output of the fast Fourier transform (FFT), can improve sensitivity in the same way as lower imaging speed does. Now on one hand, coherent averaging should be highly compatible with FDML based MHz-OCT since FDML lasers exhibit very high phase stability and the high speed effectively suppresses negative effects by sample motion [4]. On the other hand, the low transimpedance gain of MHz-OCT photodetectors increases the negative effect of non-Gaussian, correlated noises such as excess laser noise and analog to digital converter noise on the averaging result. These non-Gaussian noise contributions are more resistant to complex averaging. Hence, white noises such as shot noise or Johnson noise are not necessarily the limiting factors for effective averaging in FDML based MHz-OCT systems. A special feature of FDML based MHz-OCT is the use of optical buffering [26]. Several successive frequency sweeps, after buffering, are optical copies of the same sweep that leaves the FDML laser. Therefore, these copies hold almost the same laser noise, so that averaging of corresponding A-scans cannot effectively reduce the overall noise if the laser noise is the dominant noise source. We developed a new ultra-low noise FDML laser design with vastly reduced amplitude and phase noise. In this work we investigate the feasibility and effect of coherent averaging in MHz-OCT using this new laser.

Methods
All data shown has been acquired using a home-built OCT system including a 1300 nm FDML laser running at 411,386 Hz with optical buffer stages resulting in an eightfold effective A-scan rate of 3.3 MHz. The laser operates in the "ultra low noise" / high coherence mode and is similar to the setup described in this paper [4]. The stability of the interferometer was not specially optimized. The measured system sensitivity was 106 dB with 40 mW optical power on the sample. In one particular experimental setup the sensitivity was artificially reduced to 70 dB by inserting a neutral density filter into the sample arm. All data was acquired using a 4 GS/s, 8-bit digitizer (GaGe, CobraMax).

Coherent averaging
In complex averaging, the complex FFT output of the OCT signal is averaged. In swept source OCT, averaging the output of the FFT is equivalent to averaging the recorded transients from the photoreceiver. Direct averaging of these transients is computationally more efficient because only one FFT per A-scan is required. In the following, we will denote both averaging methods as coherent averaging and we will denote averaging of the squared magnitude as magnitude averaging.
In order to merge several A-scans into one by averaging, these should ideally be recorded at the exact same position. However, for fast scanned imaging it is usually more practical to slowly and continuously scan the sample and to use only A-scans that have the same structural information, i.e. A-scans whose spots almost entirely overlap on the sample. For the imaging experiments the translation between successive A-scans was always kept below one hundredth of the lateral resolution and can thus be assumed stationary in all images presented in this paper. The scanning step size of about one hundredth of the lateral resolution was chosen according to the results presented in section 3, which indicate less efficiency when averaging more than 100 A-scans. In practice, the number of sampled A-scans may be adjusted to the applied number of averaged A-scans to increase the imaging speed. Thus, when averaging less than 100 A-scans, the scanning step size may be increased.

Problem of timing errors in high speed swept source OCT
Timing errors are very critical in coherent averaging as they cause phase errors in the frequency domain. This specifically applies to MHz-OCT systems, since they usually operate with analog to digital converters at sampling rates of 4 Gsamples/s or more. This means, that if a timing accuracy of 1/10th of a cycle at maximum imaging depth is required, a maximum timing jitter of roughly 25 picoseconds only can be tolerated. This makes all coherent processing in fast MHz-OCT more challenging than in slower systems.

Problem of trigger jitter using high speed ADCs
Triggering the data acquisition for each A-scan is hardly possible with today's ADC-technology since electronic noise in the rising edges of the control signal for the trigger event would generate too much timing jitter. Also, the trigger rearming times of several 100 ns would cause significant data loss. To avoid trigger jitter between the recorded A-scans, the interference signals of all A-scans of a frame were recorded continuously. However, this shifts the problem of timing mismatches to the post-processing step, where the single A-scans have to be cut out from the dataset with high (sub sample interval) temporal precision. This is particularly critical at high frequencies. Close to the Nyquist limit, a shift of the interference signal by one sample interval causes a phase jump of pi, so that coherent averaging of two A-scans with such a high error would almost cause signal cancelation.

Problem of determining the laser sweep rate/sample clock with 10 −8 accuracy
Since we use one trigger event to acquire the data of an entire frame and then extract the individual A-scan data in the computer in post processing, we need to exactly know the duration of each A-scan to avoid errors in the data extraction. Assuming 1200 samples per A-scan, one B-scan consisting of 10,000 lines and a desired timing accuracy of 1/10 of a sample we need to know the ratio between the sample clock frequency of the ADC and laser sweep repetition rate with an accuracy of 1/(1200·10000·10) = 10 −8 .
So to precisely extract the A-scans of the continuously acquired data of one frame, it is necessary to know the exact ratio of the sample rate of the data acquisition and the FDML frequency of the laser, as in our case the latter is frequently adjusted by an automated control loop [27]. One way to achieve this is to synchronize the data acquisition and the FDML laser.
For example, a phase-locked loop (PLL) can be used to generate the sample clock as a multiple of the FDML frequency, given that the data acquisition card supports PLLs. This has the advantage that successive sweeps are always sampled at the same relative time positions, so that the recorded transients can be directly added for averaging. Since the data acquisition card used in this work does not allow direct clocking of the ADC and for reasons explained in the next two sections, the system was not synchronized in the way described above. Instead, the FDML frequency and the digitizing clock were generated from the same high-precision 10 MHz reference clock with known frequency ratio.

Problem of timing variations in buffered lasers
Song et al. described the problem of increased phase variation between buffered sweeps [28,29]. The time intervals between successive sweeps that are identical copies do not depend on the sample rate and FDML laser frequency, but only on the lengths of the fiber coils in the buffer stage. We accurately measured these time intervals by calculating an autocorrelation function between a laser noise dominated intensity signal (sweep of detuned FDML laser) and the signal of each of the seven copies generated in the buffer stage. Based on the maxima positions of these autocorrelation functions, it was possible to determine the different delays generated in the buffer stage with an accuracy better than 25 ps, which is ten times shorter than the resolution of the digitizer card. Since the delays are independent of the FDML frequency and the sample rate of the data acquisition, they only have to be measured once for each buffer stage. Furthermore, we expect great advantage of the precise measurements of fiber lengths or time delays for other phase sensitive OCT applications as well, as the relative phases between successive sweeps don't suffer from phase variation of the laser source. When the exact ratio of the FDML laser's filter frequency and digitizer sample rate as well as the lengths of the buffer stage coils are known, extracting the A-scans is straightforward. The original data was upsampled in the frequency domain by zero padding. After the inverse FFT, the A-scans were extracted using a linear interpolation. By this strategy it is possible to ensure correct sample timing down to the order of 25ps, effectively shifting the signal by fractional samples.
It should be noted that the buffered copies of a sweep, as they pass different fiber lengths in the buffer stages, have different time to wavelength mappings due to fiber dispersion. In this work, fiber with low dispersion around 1300 nm (Corning SMF28e) is used in the buffer stage, which makes a numerical dispersion correction before averaging unnecessary. However, this should be considered when this technique is employed using other wavelengths and fibers, e.g. FDML lasers at 1050nm for retinal imaging.

Problem of multiplexed acquisition and interleaving spurs
In modern data acquisition devices, high sampling rates are often achieved by sampling an analog input from several analog-to-digital converters (ADCs) by time division multiplexing (TDM). In the data acquisition card used in this work, four ADCs with a sampling rate of 1 GS/s are installed in order to achieve a sampling rate of 4 GS/s employing TDM. Since the different ADC channels have slightly different gain factors and offsets, even with the most accurate calibration, the A/D conversion causes distortions, so-called interleaving spurs in frequency domain. Similar effects (timing spurs) can also be caused by timing errors between the sample clocks for each ADC. In our configuration, after some magnitude averaging, the different interleaving tracks become clearly visible as noise in the vertical center of the A-scans with a maximum at half the Nyquist frequency. This results in an uneven background in the OCT images which represents the frequency-dependent average noise energy. When the digitizer card and the A-scan rate are synchronized in a way that the samples per A-scan are an integer multiple of the number of digitizers in the card, these interleaving spurs will become a phase stable fixed pattern noise that cannot be reduced by coherent averaging. However, if this situation is avoided coherent averaging can effectively reduce interleaving noise which will be illustrated in the following sections. This means in our case, that we must choose a sampling rate such that the number of samples per A-scan is not a multiple of 4.

Measurement of the averaging efficiency
In order to characterize and quantify the described problems and to investigate the effectiveness of the strategies for remedy, we performed a series of measurements without scanning on a defined mirror reflection and applied a number of different approaches to analyze the data. To estimate an upper limit for the efficiency of coherent averaging in our setup, we measured the decrease in image noise. First, an OCT dataset consisting of 1024 A-scans of a mirror was recorded with an OD 4 filter placed in the sample arm to reduce the signal intensity. The noise level was measured as a function of N averaged A-scans. Figure 1(a) shows the amplitude squared of the FFT (OCT depth signal) of a single A-scan (no averaging). The interleaving spurs are clearly visible as a narrow spectral region of increased noise around the center frequency.
Further, the mean of the amplitude squared of ten consecutive A-scans (magnitude averaged) and the amplitude squared of ten coherently averaged A-scans were calculated, displayed in Fig. 1(b) and Fig. 1(c). Comparing those signals reveals that in coherent averaging the mean noise power is significantly decreased, while the OCT signal of the mirror is similar in both averaging methods. Please note that the increase in peak amplitude of the averaged signals in Fig. 1(b) and Fig. 1(c) compared to the unaveraged A-scan in Fig. 1(a) is not a result of the averaging process. The unaveraged A-scan shown in Fig. 1(a) was randomly selected and has a lower amplitude due to intensity fluctuations caused by external influences on the imaging setup.

SNR improvement by magnitude averaging
The often-quoted statement, that magnitude averaging does not increase the signal to noise (SNR) level is not always correct. It is only true if the signal exhibits a large fluctuating background. In our case of a very stable background it can be determined by measuring a highly averaged signal prior to the actual measurement. Then this "smooth" background is subtracted from the OCT data, i.e. from the individual scans. So only the fluctuations and deviations from this highly averaged background effectively contribute to loss in SNR and noise.
As the mean noise power is frequency dependent, it causes a depth dependent background that is -in our case -almost constant from A-scan to A-scan. This background will increase the noise signal standard deviation measured within an averaged A-scan. When the mean noise power levels are very constant, as it is at least in our system, this background can be measured and subtracted and will not decrease the systems sensitivity. We use this to create a background corrected version of the 10x magnitude averaged A-scan for fair comparison of the SNR.
Therefore, this background was determined by averaging 1000 A-scans from another dataset acquired in the same way [ Fig. 1(d)]. It should be noted that this is not a real background, since it still contains the peak of the mirror reflection. The 1000x averaged "background" is subtracted from the tenfold magnitude averaged A-scan resulting in the background corrected version of the 10x magnitude averaged A-scan shown in Fig. 1(e).
To analyze the effect but also the limits of averaging, we calculated the standard deviation of the coherently averaged A-Scan [ Fig. 1(c)] and the magnitude averaged A-scan with subtracted background [ Fig. 1(e)]. Of course, for the calculation of the standard deviation only the depth signal without the reflection peak is considered. The different values of standard deviation are then plotted against the number of averaged scans, to analyze the effect of averaging on SNR improvement in both cases: coherent and magnitude averaging.

Results and discussion
Both modalities, magnitude and coherent averaging lead to a reduction in noise. However, as depicted in Fig. 1(c) and Fig. 1(e), in coherent averaging the signal amplitude is preserved while the variance of the signal is drastically reduced whereas in magnitude averaging the mean noise power is constant and only after subtracting the background "bias" the noise signal will be reduced. Theoretically, coherent averaging can achieve a noise reduction scaling with 1/N, where N is the number of averaged traces. Magnitude averaging, as known from Gaussian errors of statistically uncorrelated events, should yield a 1/sqrt(N) improvement. In the following we will discuss, how many A-scans can be averaged with our system and still achieve a noise reduction performance close to theory. We also quantify the effect of signal loss due to fringe washout by phase drift -an effect that only affects coherent averaging.  Fig. 1, which were acquired at the exact same position on the sample using a stationary beam.

Noise reduction of coherent and magnitude averaging: 1/N versus 1/sqrt(N)
The difference in noise reduction between the two versions of averaging as a function of the number (N) of averaged A-scans is illustrated in Fig. 2(a). The coherent averaging curve (blue) and the magnitude averaging curve (red) represent the mean of 80 independent measurements and the shaded area falls within one standard error of all measurements around the mean. Further, the theoretical limits are plotted, 1/N for coherent averaging (black line) and 1/sqrt(N) for magnitude averaging (dashed black line). With averaging of up to about 100 A-scans, the measured noise suppression closely follows the theoretical limit but clearly deviates when more A-scans are averaged. This shows, that averaging up to 100 A-scans is reasonable to improve the noise performance as it is almost as good as theoretically possible, whereas the gain in sensitivity when averaging more than 100 A-scans will decrease. Interestingly, for our system, this value is roughly the same for both averaging strategies.

Signal degradation due to fringe washout
In coherent signal averaging we must also avoid that the sample signal (here the peak from the reflection) is not degraded due to A-scan to A-scan phase variations. Such phase variations may arise from phase noise of the laser, revisiting errors of the scanning optics or sample movement. However, the latter also affects slower systems -an effect known as "fringe wash out" [29]. Due to the different depth-dependent contributions to the phase error described in section 2 it is very complex to quantify the total phase error value of our swept source OCT system. The problem is that some contributions to the total phase error are depth-dependent, but some are not. For example, a timing jitter or limited coherence length of the FDML laser would lead to an increased phase error value at larger imaging depth. In contrast, sample motion or related movements of the scanning optics cause a fixed phase error for all different depths. An in-depth analysis giving absolute numbers for the phase error like in [30] is beyond the scope of this paper. However, it can be easily tested whether the phase stability of the laser and interferometer is sufficient by inspecting the signal reduction caused by coherent averaging of a static sample without scanning. In our case, when using coherent averaging the maximum of the mirror signal is only decreased by −0.75 dB over 1000 consecutive A-Scans [ Fig. 2(b)], which indicates that the FDML laser as well as the electronics are phase stable enough. For magnitude averaging the peak intensity is also almost constant, the signal level only decreases by −0,5 dB [ Fig. 2(b)].

Speed reduction potential
As evident from the graph, we should be able to downscale the A-scan rate of our system by a factor of 100 yielding a sensitivity comparable to that of a shot noise limited 33 kHz OCT system. When more than 100 A-scans are averaged an increasing deviation from the theoretical limit can be observed. This indicates that further averaging becomes less effective which may be due to dominant excess or ADC related noise. It might be advisable, though, to apply incoherent averaging on coherently averaged data to decrease speckle noise, as coherent averaging has no impact on speckle reduction. All in all this means, that our FDML based 3 MHz-OCT can be effectively operated at any A-scan rate between ∼30 kHz and 3 MHz with nearly shot noise limited sensitivity.

Coherent averaging in real in vivo imaging application
To verify the benefit of coherent averaging a human finger knuckle joint was used as imaging sample. The OCT data was acquired while the finger was pressed against a cover glass and index matching gel was used to suppress reflections at the skin surface. Figure 3 illustrates the effects of coherent averaging compared to magnitude averaging. Compromising imaging speed and sensitivity, 30 consecutive A-scans were averaged in both modalities, downscaling the system speed to an effective A-scan rate of 109 kHz. Taking into account the results displayed in Fig. 2, we should theoretically be able to apply 100x coherent averaging on the OCT signal, which would correspond to an effective A-scan rate of ∼32 kHz. However, to compare the achievable image quality to ∼100 kHz OCT systems, a typical acquisition speed value of today's commercial systems (Carl Zeiss Meditec AG, ZEISS PLEX Elite 9000; Topcon, DRI OCT Triton), we only applied 30x averaging. Subsequently, 15 slightly displaced B-scans were averaged incoherently to eliminate the speckle pattern. The cut levels where set differently for the two resulting images, which is necessary for a fair comparison of the two averaging modalities since coherent averaging reduces the background instead of just better defining it as incoherent averaging does. We preferred this strategy over histogram equalization since the visual impression of the OCT signal is better comparable this way.
Besides the finger there are three faint horizontal lines visible in Fig. 3. The slightly tilted (falling) bright horizontal line at the top is the bottom surface of the cover glass. The faint slightly tilted (rising) line below is the flipped image of the top surface of the cover glass. Both lines will not disappear by coherent averaging, because they represent a "real signal". The extremely faint perfectly straight horizontal line may be caused by numerical processing and hence not be suppressed by coherent averaging.
Assessing the image quality of the finger, both averaging techniques vastly improve the image quality, and we clearly see, that coherent averaging is superior. However, the difference between the two images is not as dramatic as may be expected. While, in the case of coherent averaging, improved contrast due to the reduced uneven background can be seen, especially at large imaging depths, a visible effect of the improved sensitivity cannot clearly be claimed. Theoretically, for 30x averaging we would expect a sensitivity improvement of 10 · log(30) = 14.8 dB by coherent averaging and 10 · log √ 30 = 5 · log(30) = 7.4 dB, by incoherent averaging, a difference of 7.4 dB. The reason, why this difference does not cause a dramatic difference in image quality, may be that the sensitivity of the MHz-OCT system at 3.3 MHz A-scan rate is already very high without averaging at 106 dB.
The additional ∼15 dB of sensitivity, yielding a total of ∼121 dB for the image in Fig. 3(b) may not change the appearance of the image very dramatically, since at such high sensitivities the image quality may be limited by other parameters. Effects like multiple scattering, limited dynamic range and actually the problem to display an image with more than 70 dB dynamic on a screen while maintaining fine and detailed contrast graduation may become the limiting factors. So the usefulness of coherent averaging in OCT systems with very high sensitivity of more than 100 dB may be limited.
However, in the field of retinal OCT, MHz systems usually exhibit only around 90 dB sensitivity due to lower laser power exposure limits for the eye. In such cases, an increased sensitivity may cause a more pronounced change in image quality. To examine the effect of coherent averaging on low sensitivity systems, we reduced the sensitivity of our MHz-OCT system to 70 dB by attenuating the sample arm with an OD 1.8 neutral density filter. In Fig. 4 an unaveraged OCT image of a piece of orange as a stationary sample is compared to its 30-fold magnitude and 30-fold coherently averaged images. In this case the difference between the two averaging modalities is much more obvious. While both techniques improve the image quality, the coherently averaged image reveals more details and the uneven background, that is most distinct in the middle third of the image, is vastly reduced. Again, the cut levels were adjusted for the higher dynamic range in the coherently averaged modality. The clearly visible advantages of coherent averaging in this situation indicate that downscaling the A-scan rate might be especially useful for sensitivity limited systems. Suitable candidates might be retinal MHz-OCT systems, where the limited applicable sample arm power of 1.4 mW leads to only 90-95 dB sensitivity.

Conclusion
We investigated the feasibility of coherent averaging of A-scans using an FDML based MHz-OCT system. Due to the high electronic fringe frequencies in MHz-OCT, there are a lot of challenges to the electronics layout and the synchronization setup of the OCT-engine. We discuss step by step all the potential pitfalls and present individual solutions to them. A detailed design of our electronics-and synchronization-layout to achieve phase-stability is presented.
We show that we can average up to 100 A-scans while fully benefiting from the increased sensitivity. For averaging more than 100 A-scans this improvement deviates from the ideal curve. We expect that systems starting from lower initial sensitivity values allow for more coherent averaging since electronic noise, ADC multiplexing noise and similar, which can put a limit to the maximum achievable sensitivity, are relatively less dominant in these cases.
For our system this means that coherent computational downscaling of the 3.2 MHz-OCT to ∼30 kHz by averaging 100 A-scans is possible while fully harnessing the sensitivity improvements of slower imaging speed. We demonstrate imaging examples comparing 30x magnitude averaging with 30x coherent averaging. We observed that the improvement in image quality is rather small when using high sensitivity systems, but quite striking for low sensitivity systems.
The presented results may have a big impact on the system design of future OCT-engines. It has been realized that for many applications, a high MHz speed would be very beneficial. However, at the same time, it can be necessary to have very high sensitivity to detect faint signals -slower imaging speed might be acceptable. For example in ophthalmic OCT, MHz speed for densely sampled, ultra wide field imaging of the retina is required. Considering OCT angiography, MHz A-scan rates can improve the contrast of the human choriocapillaris and choroid as recently demonstrated [13,31]. At the same time, high sensitivity is desired to properly image the vitreous body, which usually requires lower speed due to shot noise limitations. Also, for angiographic OCT of slow flow, slower systems with higher sensitivity can be advantageous. For this reason, swept source OCT engines with adjustable sweep speed are desired. Even though most swept laser designs would in principle allow for adaptive sweep speed -even FDML lasers by adapting the buffering factor -but in reality most wavelength swept OCT lasers operate at a given frequency. In this work we prove that an OCT system with a fixed repetition rate MHz laser can perform high speed imaging with reduced sensitivity and also high sensitivity imaging at lower speed. Thus, for high sensitivity imaging slow OCT-engines are no longer needed, when the A-scan rate of high-speed systems is simply downscaled to improve sensitivity. So our results indicate that computational downscaling in software might be particularly compatible with retinal MHz-OCT systems. This will be investigated further in the future.
It should be noted, that the presented results might be different for other MHz OCT-laser sources like rapidly tunable VCSELs or stretched pulse sweeps, since they may have different noise and phase stability performance.